A Bochner Weitzenböck Decomposition for Cellular Sheaf Laplacians, Curvature and Neural Sheaf Diffusion

Francesco Maria Guadagnuolo; Alessio Borgi; Dorina Thanou; Pietro Liò

cs.CGOct 20, 2026

A Bochner Weitzenböck Decomposition for Cellular Sheaf Laplacians, Curvature and Neural Sheaf Diffusion

Authors: Francesco Maria Guadagnuolo, Alessio Borgi, Dorina Thanou, Pietro Liò

Organizations: Sapienza University of Rome, Rome, Italy · Italian Institute of Artificial Intelligence (AI4I), Turin, Italy · University of Cambridge, Cambridge, United Kingdom · ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland

Abstract

We extend the Bochner-Weitzenböck decomposition to cellular sheaf Laplacians, splitting it into a positive semidefinite term, the Bochner sheaf Laplacian, and a block diagonal term, the Forman curvature. We show that the Bochner Laplacians form a class of cellular sheaves that extends both connection and matrix-weighted Laplacians, and generalizes signed graphs to $d$ dimensions. We give a holonomy characterization of their kernel, prove the related Cheeger inequalities, and attach to them a random walk that generalizes the Gremban expansion. For the Forman term, we prove a discrete Bochner theorem, which bounds the first cohomology of a cellular sheaf with nonnegative curvature, and we read the whole decomposition as an opinion-dynamics model. We then apply this machinery to Sheaf Neural Networks, whose learnt Laplacian is otherwise unobserved. The Bochner interpretation yields a node-level conviction score that acts as an operational importance measure under frozen-model interventions, while the Forman profile provides a local descriptor of the learnt operator.

Figures & tables

Figure 1: The

\mathbb{S}^{d-1}

and

O(d)

synchronization problems on a single edge,

d=2

. (a) Each node carries a unit vector. The angular factor

W_{10}^{\top}

, here a rotation by

90^{\circ}

, transports

x_{0}

to the stalk of node

1

, and the frustration of the edge is the discrepancy between

x_{1}

and

W_{10}^{\top}x_{0}

(dashed), weighted by the radial factor

M_{10}

. The ellipse below the edge draws

M_{10}

, with principal axes labelled by its largest and smallest eigenvalues

w^{+}_{10}

and

w^{-}_{10}

: a discrepancy along the first costs more than one along the second, while the dashed circle is the isotropic case of a connection Laplacian. (b) Each node carries an orthogonal frame

g_{v}\in O(d)

, drawn as its two columns, and the frustration is

\left\lVert g_{1}-W_{10}^{\top}g_{0}\right\rVert_{M_{10},F}

Figure 2: The random walk of a Bochner matrix as a generalized Gremban expansion. (a) For the signed triangle with

w_{01}=-3

(dashed) and

w_{02}=1

, the Gremban expansion doubles each node into

v^{+}

and

v^{-}

. From node

0

the walker moves to node

1

with probability

3/4

and to node

2

with probability

1/4

, in proportion to

|w_{0u}|

; crossing the negative edge flips its sign,

0^{\pm}\to 1^{\mp}

, and crossing the positive one keeps it,

0^{\pm}\to 2^{\pm}

. (b) For

d>1

each node is lifted to a sphere

\mathbb{S}^{d-1}

, and the jump probability

p_{0u}(\hat{x})=\hat{x}^{\top}P_{0u}\hat{x}

depends on the direction the walker carries: in this example, from

y

it reaches nodes

1

and

2

with probabilities

3/4

and

1/4

, from

x

with

1/2

each (Definition 4.3 ).

Figure 3: The three examples of this section, with

d=1

. (a) The empty triangle of Example 5.1 : each arrow carries the restriction map from a vertex to an edge, all equal to

1

except

\mathcal{F}_{0\trianglelefteq[0,2]}=\mathcal{F}_{1\trianglelefteq[1,2]}=2

. (b) The star

S_{4}

of Example 5.2 , with maps

1

from the center and

A

from the leaves. (c) The heptagonal bipyramid of Example 5.3 seen from above: the vertices

A

and

B

, one on each side of the equator, project onto the same point at the center. Next to each vertex is its value of

q

, which sets the restriction maps of the construction.

Zero masking	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$\mathbf{0.51}_{\pm 0.17}$	$\mathbf{1.34}_{\pm 0.39}$	$\mathbf{2.18}_{\pm 0.70}$	$\mathbf{3.53}_{\pm 1.26}$	$\mathbf{4.55}_{\pm 1.83}$
Low conviction	$\mathbf{0.02}_{\pm 0.01}$	$\mathbf{0.04}_{\pm 0.04}$	$\mathbf{0.07}_{\pm 0.07}$	$\mathbf{0.20}_{\pm 0.14}$	$\mathbf{0.39}_{\pm 0.32}$
High degree	$0.37_{\pm 0.16}$	$0.75_{\pm 0.28}$	$1.06_{\pm 0.40}$	$1.76_{\pm 0.68}$	$2.46_{\pm 0.98}$
Low degree	$0.03_{\pm 0.02}$	$0.19_{\pm 0.10}$	$0.38_{\pm 0.21}$	$0.85_{\pm 0.44}$	$1.35_{\pm 0.74}$
Degree-matched (high)	$0.06_{\pm 0.07}$	$0.20_{\pm 0.17}$	$0.38_{\pm 0.32}$	$0.57_{\pm 0.51}$	$0.93_{\pm 0.62}$

Table 1: Relative drop in performance (%) under zero masking, by selection strategy and budget. Mean

{}_{\pm\text{s.e.}}

across 8 datasets, each averaged over its folds. In each column, the most damaging and the least damaging selection are highlighted.

Panel of Table 2 — Table 2: $1$ -Wasserstein distance between the Forman profile of the learnt normalized GeneralSheaf and the topological reference, by training epoch (rows; $0$ is the untrained sheaf) and relative depth of the layer (columns). Darker cells are farther from the reference: the distance shrinks with training and with depth. Mean ${}_{\pm\text{s.e.}}$ across 10 datasets, each averaged over its folds; 8 for the middle layer, which networks with $L=2$ lack.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

	Checkpoint
Observable	Untrained	Best epoch
Spearman $\rho$ with degree	$0.07_{\pm 0.10}$	$0.33_{\pm 0.07}$
Spearman $\rho$ with betweenness	$0.05_{\pm 0.08}$	$0.28_{\pm 0.06}$
Spearman $\rho$ with confidence	$0.17_{\pm 0.07}$	$0.66_{\pm 0.05}$
Spearman $\rho$ with saliency	$0.07_{\pm 0.10}$	$-0.18_{\pm 0.09}$
$\|$ Cohen’s $d\|$ , train split	$0.21_{\pm 0.05}$	$0.54_{\pm 0.09}$

Appendix

Table 3: Spearman correlation between the conviction and four node properties, and absolute Cohen’s

d

between the conviction of correctly and incorrectly classified nodes, at the last diffusion layer. Untrained is the epoch-

0

checkpoint, a control.

Mean masking	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$\mathbf{0.50}_{\pm 0.16}$	$\mathbf{1.34}_{\pm 0.39}$	$\mathbf{2.19}_{\pm 0.72}$	$\mathbf{3.61}_{\pm 1.30}$	$\mathbf{4.66}_{\pm 1.85}$
Low conviction	$\mathbf{0.01}_{\pm 0.02}$	$\mathbf{0.08}_{\pm 0.05}$	$\mathbf{0.14}_{\pm 0.10}$	$\mathbf{0.36}_{\pm 0.17}$	$\mathbf{0.57}_{\pm 0.38}$
High degree	$0.38_{\pm 0.16}$	$0.94_{\pm 0.38}$	$1.35_{\pm 0.59}$	$2.13_{\pm 0.94}$	$2.91_{\pm 1.17}$
Low degree	$0.03_{\pm 0.02}$	$0.18_{\pm 0.11}$	$0.41_{\pm 0.22}$	$0.88_{\pm 0.48}$	$1.43_{\pm 0.80}$
Degree-matched (high)	$0.09_{\pm 0.07}$	$0.30_{\pm 0.21}$	$0.52_{\pm 0.37}$	$0.77_{\pm 0.63}$	$1.21_{\pm 0.71}$

Appendix

Table 4: Budget sweep under mean masking, laid out as Table 1 . High conviction is the most damaging selection at every budget, low conviction the least.

Hard gating	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$\mathbf{0.81}_{\pm 0.29}$	$\mathbf{1.80}_{\pm 0.41}$	$\mathbf{2.77}_{\pm 0.66}$	$\mathbf{4.32}_{\pm 1.27}$	$\mathbf{5.67}_{\pm 1.89}$
Low conviction	$0.05_{\pm 0.03}$	$\mathbf{0.11}_{\pm 0.04}$	$\mathbf{0.18}_{\pm 0.09}$	$\mathbf{0.43}_{\pm 0.22}$	$\mathbf{0.71}_{\pm 0.45}$
High degree	$0.65_{\pm 0.20}$	$1.37_{\pm 0.37}$	$1.91_{\pm 0.45}$	$2.83_{\pm 0.73}$	$3.67_{\pm 1.04}$
Low degree	$\mathbf{0.05}_{\pm 0.02}$	$0.26_{\pm 0.12}$	$0.58_{\pm 0.27}$	$1.20_{\pm 0.59}$	$1.91_{\pm 0.98}$
Degree-matched (high)	$0.12_{\pm 0.05}$	$0.37_{\pm 0.19}$	$0.68_{\pm 0.35}$	$1.02_{\pm 0.60}$	$1.44_{\pm 0.86}$

Appendix

Table 5: As Table 4 , under hard gating. At the

1\%

budget low degree is marginally the least damaging.

Pruning	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$\mathbf{1.65}_{\pm 0.96}$	$\mathbf{2.95}_{\pm 1.10}$	$\mathbf{4.02}_{\pm 1.16}$	$\mathbf{5.64}_{\pm 1.49}$	$\mathbf{7.08}_{\pm 2.04}$
Low conviction	$\mathbf{0.01}_{\pm 0.01}$	$\mathbf{0.07}_{\pm 0.05}$	$\mathbf{0.16}_{\pm 0.11}$	$\mathbf{0.42}_{\pm 0.28}$	$\mathbf{0.81}_{\pm 0.57}$
High degree	$1.36_{\pm 0.77}$	$2.68_{\pm 1.16}$	$3.48_{\pm 1.24}$	$4.82_{\pm 1.38}$	$5.77_{\pm 1.57}$
Low degree	$0.04_{\pm 0.03}$	$0.24_{\pm 0.14}$	$0.54_{\pm 0.30}$	$1.13_{\pm 0.67}$	$1.88_{\pm 1.10}$
Degree-matched (high)	$0.20_{\pm 0.05}$	$0.52_{\pm 0.21}$	$0.89_{\pm 0.41}$	$1.29_{\pm 0.76}$	$1.73_{\pm 1.09}$

Appendix

Table 6: As Table 4 , under pruning.

Zero masking	Ranking criterion
Bin	Conviction	Confidence	Degree
1	$0.07_{\pm 0.07}$	$0.35_{\pm 0.21}$	$0.38_{\pm 0.21}$
2	$0.09_{\pm 0.08}$	$0.36_{\pm 0.21}$	$0.42_{\pm 0.22}$
3	$0.06_{\pm 0.20}$	$0.39_{\pm 0.26}$	$0.41_{\pm 0.24}$
4	$0.17_{\pm 0.25}$	$0.42_{\pm 0.25}$	$0.37_{\pm 0.27}$
5	$0.37_{\pm 0.26}$	$0.48_{\pm 0.38}$	$0.45_{\pm 0.29}$

Appendix

Table 7: Decile experiment under zero masking. The training pool is split into ten equal bins by each ranking criterion (columns), from the lowest-scoring bin

1

to the highest-scoring bin

10

(rows), and one bin is perturbed at a time. The conviction has the largest drop at bin

10

, and most of its growth occurs in the last two bins.

Zero masking, avg_z	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$\mathbf{0.56}_{\pm 0.17}$	$\mathbf{1.68}_{\pm 0.57}$	$\mathbf{2.65}_{\pm 0.98}$	$\mathbf{4.10}_{\pm 1.61}$	$\mathbf{5.18}_{\pm 2.21}$
Low conviction	$\mathbf{0.01}_{\pm 0.01}$	$\mathbf{0.02}_{\pm 0.03}$	$\mathbf{0.02}_{\pm 0.06}$	$\mathbf{0.13}_{\pm 0.18}$	$\mathbf{0.34}_{\pm 0.29}$
High degree	$0.37_{\pm 0.16}$	$0.75_{\pm 0.28}$	$1.06_{\pm 0.40}$	$1.76_{\pm 0.69}$	$2.46_{\pm 0.98}$
Low degree	$0.03_{\pm 0.02}$	$0.19_{\pm 0.10}$	$0.38_{\pm 0.21}$	$0.85_{\pm 0.44}$	$1.35_{\pm 0.74}$
Degree-matched (high)	$0.07_{\pm 0.04}$	$0.19_{\pm 0.15}$	$0.27_{\pm 0.26}$	$0.52_{\pm 0.49}$	$0.77_{\pm 0.55}$

Appendix

Table 8: As Table 1 , with nodes ranked by the layer-averaged score avg_z . The effect is unchanged, and slightly sharper.

Ranking score	Relative drop (%)
Soft gating
Confidence	$0.67_{\pm 0.37}$
Conviction	$3.09_{\pm 0.95}$
Degree	$4.19_{\pm 1.95}$

Appendix

Table 9: Relative drop under soft gating, which attenuates the outgoing message of every training node in proportion to its score (gate floor

g_{\min}=0.1

); with no budget, each ranking score gives one number.

Zero masking, high		Budget (fraction of the training pool)
Dataset	Selection	1%	5%	10%	20%	30%
Amazon Ratings	Degree	$1.25_{\pm 0.17}$	$1.75_{\pm 0.28}$	$2.14_{\pm 0.32}$	$3.14_{\pm 0.32}$	$3.90_{\pm 0.27}$
	Conviction	$1.16_{\pm 0.17}$	$2.31_{\pm 0.26}$	$2.93_{\pm 0.29}$	$3.81_{\pm 0.28}$	$4.48_{\pm 0.28}$
Citeseer	Degree	$-0.04_{\pm 0.16}$	$-0.39_{\pm 0.21}$	$-0.65_{\pm 0.35}$	$-1.25_{\pm 0.82}$	$-1.74_{\pm 1.05}$
	Conviction	$0.06_{\pm 0.17}$	$0.41_{\pm 0.21}$	$0.62_{\pm 0.22}$	$1.20_{\pm 0.31}$	$0.67_{\pm 0.46}$
Cora	Degree	$0.46_{\pm 0.17}$	$0.87_{\pm 0.20}$	$0.99_{\pm 0.37}$	$1.53_{\pm 0.44}$	$2.21_{\pm 0.45}$

Appendix

Table 10: Budget sweep under zero masking, dataset by dataset, for the high-degree and high-conviction selections. Unlike the other tables, the spread is the standard error across the ten folds of each dataset.

Zero masking, low		Budget (fraction of the training pool)
Dataset	Selection	1%	5%	10%	20%	30%
Amazon Ratings	Degree	$0.02_{\pm 0.05}$	$-0.02_{\pm 0.08}$	$0.05_{\pm 0.12}$	$0.48_{\pm 0.18}$	$0.73_{\pm 0.20}$
	Conviction	$-0.02_{\pm 0.03}$	$-0.02_{\pm 0.08}$	$0.07_{\pm 0.09}$	$0.13_{\pm 0.10}$	$0.56_{\pm 0.22}$
Citeseer	Degree	$-0.04_{\pm 0.04}$	$-0.05_{\pm 0.11}$	$-0.12_{\pm 0.16}$	$-0.15_{\pm 0.22}$	$-0.29_{\pm 0.35}$
	Conviction	$0.08_{\pm 0.09}$	$-0.12_{\pm 0.18}$	$-0.30_{\pm 0.20}$	$-0.62_{\pm 0.27}$	$-1.60_{\pm 0.67}$
Cora	Degree	$0.00_{\pm 0.00}$	$0.09_{\pm 0.06}$	$0.09_{\pm 0.14}$	$0.11_{\pm 0.19}$	$0.14_{\pm 0.18}$

Appendix

Table 11: As Table 10 , for the low-degree and low-conviction selections.

Pruning	Budget (fraction of the training pool)
Selection	1%	5%	10%	20%	30%
High conviction	$7.7_{\pm 3.1}$	$15.9_{\pm 3.5}$	$22.0_{\pm 3.4}$	$31.3_{\pm 3.1}$	$39.4_{\pm 2.8}$
Low conviction	$0.5_{\pm 0.1}$	$2.7_{\pm 0.5}$	$5.6_{\pm 1.0}$	$11.6_{\pm 1.8}$	$18.0_{\pm 2.5}$
High degree	$10.2_{\pm 3.4}$	$23.5_{\pm 5.1}$	$32.7_{\pm 5.5}$	$44.2_{\pm 5.1}$	$51.6_{\pm 4.3}$
Low degree	$0.4_{\pm 0.1}$	$1.9_{\pm 0.6}$	$4.2_{\pm 1.2}$	$8.7_{\pm 2.5}$	$13.3_{\pm 3.7}$
Degree-matched (high)	$5.0_{\pm 1.3}$	$11.5_{\pm 1.4}$	$16.6_{\pm 1.3}$	$23.6_{\pm 1.1}$	$28.3_{\pm 1.7}$

Appendix

Table 12: Percentage of the graph’s edges deleted under pruning, by selection strategy and budget; the performance cost of the same selections is in Table 6 . Selecting by degree deletes the most edges.

Panel of Table 13 — Table 13: $W_{1}$ distance of the unnormalized GeneralSheaf from the topological reference (top) and the zero reference (bottom). The profile aligns with neither.

Panel of Table 14 — Table 14: $W_{1}$ distance of the other Laplacian, with maps trained unnormalized and evaluated normalized (Appendix D.2 ), from the topological reference. Once normalized, the maps align with it as training proceeds.

Panel of Table 15 — Table 15: $W_{1}$ distance of the other Laplacian, with maps trained normalized and evaluated unnormalized (Appendix D.2 ), from the topological reference (top) and the zero reference (bottom). The maps lose their alignment and stay far from the zero reference.

Panel of Table 16 — Table 16: $W_{1}$ distance between the two Laplacians evaluated normalized, built from the maps trained normalized and unnormalized (cross-comparison, Appendix D.2 ). They stay apart and neither converge nor diverge.

Panel of Table 17 — Table 17: As Table 16 , with both Laplacians evaluated unnormalized. The distances are of the same size.

Panel of Table 18 — Table 18: $W_{1}$ distance of the unnormalized JointSheafParams, started at the identity sheaf, from the topological reference (top) and the zero reference (bottom), the latter measuring the deviation from the identity. The maps deviate through the layers and are drawn back with training.

Panel of Table 19 — Table 19: $W_{1}$ distance of the normalized JointSheafParams, started at the identity sheaf, from the topological reference, which here measures the deviation from the identity. The deviation grows with training. Tolokers is excluded since its training is unstable ( $9$ datasets, $7$ for the middle layer). On Cora, PubMed and Questions training was interrupted, and these datasets enter with the folds that completed (between two and six of ten).

Panel of Table 20 — Table 20: Normalized GeneralSheaf on Roman Empire, $d=5$ , five layers, against the topological reference.

Panel of Table 21 — Table 21: Unnormalized GeneralSheaf on Citeseer, $d=5$ , four layers, against the topological and the zero reference.

Panel of Table 22 — Table 22: Unnormalized JointSheafParams on Questions, $d=5$ , four layers, started at the identity sheaf, against the topological and the zero reference. In the first layer the profile is constant, so its correlation is undefined.

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Sheaf neural networks generalize graph learning with local vector spaces and edge maps that model how information is compared and transported.

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We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a restriction map on an edge. The restricted coboundary operator on the free coordinates is unitriangular, so its determinant is

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